Energy growth in Hagen-Poiseuille flow of Herschel-Bulkley fluid

Linear stability of Poiseuille flow of Herschel-Bulkley fluid in a cylindrical pipe is studied using modal and non-modal approaches. The first part of the present study thus deals with the classical normal mode approach in which the resulting eigenvalue problem is solved using a Chebyshev collocation method. Within the considered range of parameters, the modal-linear theory predicts that perturbations are dumped exponentially. In the second part, the effect of the rheological behavior of the fluid on the pseudospectra and the most amplified perturbations is investigated. At very low Herschel-Bulkley number (Hb < <1), the optimal perturbation consists of almost streamwise vortices, and the amplification of the kinetic energy is provided by the lift-up mechanism. In contrast, for sufficiently large values of Hb, the optimal perturbation is axisymmetric and the growth of the kinetic energy is provided by the Orr-mechanism. For intermediate values of Hb, the optimal perturbation is oblique. The amplification of such perturbation is due to a synergy between Orr and lift-up mechanisms. In the last part of the study, the maximal value of the Reynolds number, RecE, below which the perturbation energy decreases monotonically with time is computed for a large range of Hb. Asymptotic behaviors of RecE for Hb < <1 and Hb > >1 are established. The influence of the terms arising from the viscosity perturbation is highlighted throughout this study.